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This clip is from the movie Holes, which is inexplicably billed as a movie for kids. (Sue Van Hattum kindly brought it to my attention.) The grim premise is a penal colony of children, each digging one hole per day in the desert for the duration of their sentences. On our hero Stanley Yelnats’ first day, he accidentally takes another kid’s shovel which is slightly shorter than the rest. Drama!

Steve: “How long will it take Dan to go up the down escalator?” +15 othersOther questions. +2

It’s obvious to me which problem has the stronger current. Maybe I can do something about that; maybe I can’t. Regardless, I had to make a more authoritative call on the problem than I prefer. I said, “Okay, let’s talk about the first question. ‘Does it really matter that X-Ray’s shovel is a couple inches shorter?’

“How many pounds of extra dirt is Stanley going to dig at the end of a full year?”

The movie doesn’t define the shorter shovel’s length, which leads to an awesome moment where the students and the teacher can basically make something up, some number that has no material effect whatsoever on the mathematics they’re practicing but which gives everyone the sense that “this is our problem.” Big win.

Director Andrew Davis didn’t think it fit the narrative of his film to mention the weight of a cubic foot of desert dirt so we faced a similar dilemma w/r/t density.

Step Four:

Give them a pile of information to use as they see fit.

Step Five:

Give them time to work.

I put twenty minutes on the clock and asked everyone to email me either a scan or a camera photo of their work when they finished.

After they compute their final answer, ask them to compare it to their error bounds from step two.

Play the answer video.

Compare the answer to our guesses from step two. Determine who guessed closest.

Discuss sources of error.

Discuss follow-up questions.

Stacy: I love this problem, but we still don’t have a good way to check our answer.

So this problem receives a certain demerit for not allowing us to observe the answer. I accept that demerit.

These problems require some kind of plan for challenging students who finish early. The attendees offered two approaches I want to highlight here:

Aaron: change the size of the original shovel.Justin & Anna: how much shorter would the next shovel have to be for the difference to be the same?

Aaron has changed the input quantities and asked his students to find another output. His students will use the same operations on different numbers. From my experience, this leaves the teacher vulnerable to charges of assigning busy work.

Justin and Anna, by contrast, have made the old output quantity the new input. Before, their students were solving for the total quantity of dirt. Now, Justin and Anna have given their students the quantity of dirt and asked them to work backwards to new inputs. This is a great, versatile way to quickly create a new problem for students who finish early.

Open Questions

ft3 or pounds? In a workshop recently, we ended this problem by calling a local composting company and asking them how many cubic feet of compost they brought in on each dumptruck. The units on our final answer, then, were “dumptrucks.” There are different, subtle ways to frame the same question. Do you ask for mass, weight, volume, or dumptrucks? Mr. K and I went back and forth over the difference between these two questions. “How many extra pounds of dirt will Stanley dig after a year?” vs. “How many extra cubic feet of dirt will Stanley dig after a year?” What are their advantages and disadvantages? I’m still going over this in my head but my sense is that our students’ early estimates will be more accurate in ft3, but the answer will be more tangible to them in pounds. (See also: How Big Really?.)

I’m curious, if you were in that session, what does it do for your engagement with the problem to see your classmates’ guesses? (I welcome any other comments from the participants, of course.)

What app will let people email files to a public folder? I planned to use Dropbox but, last minute, I realized it didn’t support that function so I had participants email me their files, which I uploaded to my domain after updating an HTML file, and the whole thing was ridiculous. How do I cut out the middleman (me) here?

I had DimDim’s whiteboard up at the start of the workshop, which turned out to be accidentally awesome. Participants started doodling as they waited for the session to start. One participant drew a map of the US and asked everyone to identify their location with a star.

I need to type questions after I speak them so the responses in the transcript make a little more sense to me afterwards.

I’m crazy enough to look up the shooting locations for Holes. It remains to be seen if I’m crazy enough to drive down to the Mojave Desert with a scale and weigh a cubic foot of dirt, which is clearly what needs to happen here.

55 Responses to “WCYDWT: Dirt”

Why Email the files to a public folder? You could just register a Photobucket account and give everyone the username/password and we can directly upload it.

I still think cubic ft works better because a cubic foot is a cubic foot. End of story. We don’t use such units often enough to make it completely intuitive, but we use it far more than the weight of a cubic foot of dirt I’m sure.

I like Jon’s point about “a cubic foot is a cubic foot”. And you could help the kids understand what their answer means by, for example, determining how tall a rectangular-prism-shaped building would have to be to equal that number of cubic feet. (The class can determine what reasonable dimensions for the base might be.) If the answer turns out to be as ridiculously tall as the number of pounds turned out to be huge, that could be pretty dramatic. I imagine height of a building would be much more intuitive for students than volume.

You’re probably looking at sandy loam, which is 1.44 g/cm^3
but any number from 1.25 to 1.5 g/cm^3 is reasonable.

Incidentally, this sort of problem makes it clear why metric is easier: I can remember that water is 1 g/cm^3 or 1 pound/pint, but the conversion between cubic feet and gallons always escapes me. I have no such trouble with cm^3, liters, and m^3.

I’d do the whole problem in metric units and just convert kg to pounds at the end.

Very enjoyable DimDim session the other day. As for a follow up, I found that seeing other people’s guesses was great. Doing so made me question my initial guesses and think about who might have the best guess.

I also think that some of the mentioned downfalls of this problem could actually be positives. Without a certain answer or a clear question there is the opportunity for students to form many different problems. This could lead to a discussion of which problems are best and which we can answer with the information given in the video. I think this process can be beneficial to students developing problem solving skills. With that in mind this might not be the first WCYDWT lesson I use but maybe one I use later in the year when students are more comfortable with the process and confident in their work.

Also, I was wondering if anybody had any multimedia software recommendations for a PC to make movies such as the one featured here. Any suggestions would be great.

There is another factor to be considered here, energy expended. The weight of the dirt and the energy expended to lift it is the easy end of the process, breaking the dirt loose into a shovelable form requires more energy than simply lifting the loose dirt. Factor in rocks, dry compact soil at depth, dust and heat, and the size of the hole becomes more of a concern than weight of or amount of dirt. Digging holes in the desert isn’t easy and here real life experience plays a role coming up with the best solution to the problem.

I had problems with audio at the beginning of the session, so missed the part where you asked us all to give our best guess. I’m curious if the other participants felt like Peter, in that seeing other guesses made them question their own.

As Peter mentioned, because there is no final answer, this might play out better later in the year. Especially if the students have become accustomed to seeing the answer video. It might really drive some of them crazy not to know for sure, but this could lead to some interesting questions and discussion.

I’d like to see how this question goes with students just considering cubic feet. As Steve G. mentioned, it is probably easier to visualize volume as opposed to tons. “If we had a container whose base was the area of our classroom, how high would all the extra dirt pile up that Stanley shoveled?”

As for file management, how about have students set up their own blog/website at the beginning of the year and they can upload their files, write comments, discuss via posts there.

Then you can just use your favorite feed reader to create a list of RSS feeds to all their sites. Puts all the work on them to maintain their pages and you get automatically updated on their assignments in real time (including date/time posted if that’s important to you). And it’s hosted on their site instead of clogging up your email.

Plus, they end up with a portfolio of work at the end that can be saved and used for variety of reflections, assessments, resources, etc.

I love weebly.com for this kind of stuff. Blogger would be simple to set up too. Weebly also has educational sites (education.weebly.com) where you can create one page that has a forum and allows for a file management system where students can upload files that are directly sent to you.

Great stuff…gonna try this Holes one w/ my middle schoolers this year.

I was just looking at the four submitted answers. Everyone assumed a cylindrical hole. I don’t remember from the book (or the flick) if a hemispherical hole was a possibility. I’ll turn one of my children on the task today as they wind up their summer vacation this week.

A challenge question: What is the difference in the amount of dirt between the two type of holes?

BTW: The twist on why the kids had to dig the holes is great. If anyone has not read this book, you should! The movie just doesn’t cut it. It is a three – four hour read max!

Am I misreading the problem or are you all missing the point of the shorter shovel? In the movie (and book, I assume), the hole has to be equal to the diameter and depth (if memory serves) of a shovel. Hence, a shorter shovel SAVES digging, not the other way around.

p.s.: I tried participating on Saturday but got there a little late. Couldn’t figure out how to track back to where things started and hence couldn’t answer the questions you were asking, Dan. Am I alone in having that problem? I didn’t have the information to answer what was being asked. Maybe there’s a way to accommodate late-comers?

Maybe this is obvious, but the actual, real world difference is that, for the whole day while you’re digging, you think “Hey, at least I have the short shovel”. That’s why it would be worth fighting over. The calculation, if it comes out favorably, is how you console yourself when you don’t get it (“Well, it only means this many extra shovelfuls. It’s not as big a deal as they make it out to be.”).

Pursuant to that version of things, the correct units are shovelfuls and the time period is probably per day, since you do your time a day at a time, and try not to think about the long haul.

Other peoples guesses are always nice because it creates competition which creates desire to solve the problem. Though not every student is competitive.

I would say you wouldn’t have to go to the Mojave but just pick a pretty dry area near you to weigh the dirt. It would be cool to have a ft^3 box and bring it in full of dirt too. Then the students could pick it up and see the size so both units would be more tangible. I’m liking this more and more.

Jon Voisey: Why Email the files to a public folder? You could just register a Photobucket account and give everyone the username/password and we can directly upload it.

Because sending an e-mail is easier for most people than logging into a new utility and uploading a file. Plus I got a bunch of PDFs, which Photobucket doesn’t support. At this point I have a good line on drop.io (from Dan Anderson) and Evernote (from Bud Hunt).

Claire: If we had a container whose base was the area of our classroom, how high would all the extra dirt pile up that Stanley shoveled?

I really like this. It’s tangible to me and descriptive in a way that cubic feet doesn’t match while being easier to estimate than pounds. I’m guessing the range of guesses will be a lot tighter (like 10 feet to 100 feet) than pounds (which had a range of something like 5,000).

MPG: Hence, a shorter shovel SAVES digging, not the other way around.

Hence the question, “How many pounds of extra dirt is Stanley going to dig at the end of a full year?” I think we’re all on the same page here.

msouth: the correct units are shovelfuls

Any plan for measuring that?

Mr. K: If you’re willing to wait until the end of October, I can do that for you.

In Chapter 7 Stanley is narrating: “…X-Ray claimed it was shorter than all of the others (5-foot) but if it as, it was only by a fraction of an inch” and later “… Stanley’s hole had to be as wide and deep as his shovel and it had to lie flat on the bottom of the hole in any direction”. Looks like a cylindrical hole is the only way to go.

Here are some thoughts I’ve had about the problem and the session over the intervening days.

With regard to the units used for the final answer, I think “holes” might be a natural and compelling unit to use. I’m not sure how to phrase the question, but something like, “By using the shorter shovel, X-Ray avoids digging the equivalent of how many full-sized holes over the course of a year?” With a different tack, maybe the question, “How much less work does X-Ray do in a year?” would allow for “holes” to be suggested as a unit, or perhaps different students would then actually opt for a variety of different units. The latter situation would be interesting because then there would have to be conversions made in order to compare hard-fought-for-and-invested-in answers.

Using full-sized holes as a unit gives a different feel to the result—the immensity of pounds or cubic feet is stunning, but the number of holes ties the quantity more closely to the narrative context. A bunch of pounds means a lot of work, sure, but it’s clear from the start that everyone is digging a lot of dirt. The fact that the shorter shovel could save X-Ray from digging the equivalent of 35 holes in a year—more than a month’s work—makes for some compelling narrative.

Note that asking it in this way removes the need for finding out the density of dirt. For better and worse.

It also allows for a nice geometrical connection to be made, namely, to scale factor. From this perspective, even the shape and size of the original hole are irrelevant. The linear measurements of the smaller hole are scaled by 29/30 from the full-sized hole, and so its volume is scaled down by that cubed, which means that the smaller hole has about 90.3% of the volume of the original one. So X-Ray is digging only 90.3% as much, and that’s the same as digging only 90.3% of the days: 330 full days, 330 full-sized holes.

As far as my experience of some aspects of the session: I didn’t feel like I had enough time to really process others’ estimates. I think that asking for revised estimates (or a consensus estimate) once everyone has seen everyone else’s guesses could be a good strategy, both for drawing attention to the activity of estimation and provoking some critical thinking with regard to revision.

Having a smoother way to share our written work will be nice.

I agree that typing in your questions (in addition to saying them aloud) is a good idea. You might also be able to find a way to use the whiteboard space to help guide the flow of conversation and to highlight interesting points that arise. (Maybe stick the estimates up there?)

The whole experience felt a little rushed, but I think that was mostly just me getting used to the format and the technology. I felt like having time to watch the video twice might be important. It was nice that we had some time to debrief and toss some questions and opinions around at the end.

Thanks for hosting, Dan, and for demonstrating the potential of online PLNs going “live”!

Thanks for the problem I used it today as my opener in all of my classes. We never got into the soil density, but had great conversations about conversions and volume. My students liked to just talk in terms of how many more holes would be dug per inch of shovel per year. I haven’t worked with the problem too much myself as I do not want to give any help and I find that if I do not work the problem ahead of time I tend to help less, but most were talking that 2 inches less in shovel length meant around 32 less holes dug. That is from informally surveying them. Great problem thanks for the work. It made a great intro on my first day of the new year. And worked great to emphasize showing work, labeling everything and explaining what they did.

I think digging hemispherical holes is more common, although I’m not sure “typical” is the name of the game at Camp Green Lake (yes, I loved the book, too!). Good point there.

@ Justin:

I love the idea of “holes saved per year.” That makes it much more accessible (though not necessarily easier), which is important to people who get psyched out by math (me!).

Finally, maybe I’m a weirdo….But before I even thought of the size of the hole, I thought about whether it’s easier to dig with a shorter or longer shovel. That’s the thought that flitted into my mind when I read the book, with no mathematical context (i.e. me thinking “there’s a math question in this somewhere…”). That has to do with energy expended rather than amount of dirt. My brief encounters with high school physics don’t give me enough background knowledge to deal with that issue…but maybe it would put a good spin on it for a physics class?

I like the idea of being able to see the guesses that everyone made. However, looking through ~20*3 numbers is hard (especially when the numbers are big). What about asking people to submit their [low, estimate, high] values via a Google form? And then it seems you could create a quick/simple visualization of all the guesses, which could later be updated with the correct answer to compare the different guesses.

About using guessing as part of activity. I find it really enhances the lesson. I do it mostly with middle schoolers and they like the idea of seeing who had the best guess at the end – especially if there is a prize. The other neat thing I discovered is that sometimes the average of their guesses is better any particular guess! Or the opposite can happen. If the problem results in a surprise answer i.e. the birthday problem – the guesses for two people sharing the same birthday in a given classroom of 23 students is much lower than the 50-50 result.

“You’re probably looking at sandy loam, which is 1.44 g/cm^3
but any number from 1.25 to 1.5 g/cm^3 is reasonable.

Incidentally, this sort of problem makes it clear why metric is easier: I can remember that water is 1 g/cm^3 or 1 pound/pint, but the conversion between cubic feet and gallons always escapes me. I have no such trouble with cm^3, liters, and m^3.

I’d do the whole problem in metric units and just convert kg to pounds at the end.”

– Funniest thing EVER. If the kids actually get this caught up in it, someone should make a movie.

In regards to session feedback, I found the program easy to use but any online chat system makes me wary as the fastest typer tends to win. The ‘type in your answer and hit enter why I say’ portion was fabulous and I wish I could find a better way to do that in a classroom that made sense other than giving them my cell number and having them text me their thoughts. Though that is an interesting idea now that I’ve typed it, and one that might work if I can ensure at least one phone per table with unlimited texting (likely for my school).

I was surprised at how hesitant I was to put up a guess. I have good spacial ability, but not in terms of quantity. In class I would probably have students write down their guesses on slips that I could collect and toss up on the board while they worked on a part of the problem.

Fixing the final answer in a unit the students can see and make sense of (Like Claire’s classroom or # of extra holes) is an awesome idea and perhaps something each class could decide on individually. Makes me what to have some data on cubic feet dimensions for things like the gym at my school as that is something all students can see in their minds. Might need to get a bigger measuring tape… Pounds I’m more leery of as I don’t think most people have a good grasp of them. The very fact that 20+ adults didn’t have a good estimate for 1 ft^3 of dirt proves that to me.

My last thought is on the variety of questions this clip drove people toward. It’s a reminder to not live in the bubble of one’s own classroom and that sharing ideas with other teachers (even if that means a bit of stalking them after school) is critical for making well-though out lessons. I doubt I would have thought of some of the great ideas thrown out by the other people in the session (high-fives to Justin & Anna as I particularly liked their follow up question about next shovel length).

I love this problem. I think that the lack of an “answer video” is a good opportunity to discuss the fact that, in real life, you can’t flip to the back of the book for the answers. The mathematician must think of ways to verify his/her answer – and one way to do that is to collaborate with others.

If you want to have a closing video, you could go to a place that sells dirt and take a video of yourself with a pile of dirt that is roughly equivalent to the amount of extra dirt that Stanley has to shovel. (It would be funny to video yourself shoveling the pile and then pan out to show how big the pile really is.)

I also like the idea of envisioning how deeply the extra dirt would fill the classroom and also bringing in a cubic foot of dirt in a container. I had no concept of how heavy that container would be.

I was confused about who’s who with the shorter shovel, where’s it shorter (shaft, blade, …?) and why does it matter.

To quote the video: “you dig a hole every day, five foot deep, five foot diameter, _your shovel is your measuring stick_”.

Then, Mr. Goldenberg says “Hence, a shorter shovel SAVES digging, not the other way around.” in response to the problem statement, “[Stanley takes another kid’s shovel which is slightly shorter … ] How many pounds of extra dirt is Stanley going to dig at the end of a full year?”

I think the problem is being inconsistently stated; if X-ray has the longer shovel, he’s the guy digging more. If Stanley has the longer shovel, the the set-up is wrong. Something inconsistent is going on here. But if you swap one of the variables, it all comes out right (or just let the answer be negative).

Also, one might also state the question more openly: “How _much_ extra dirt [is being dug by Mr. Röntgen or whomever drew the shorter straw or shovel or something]”, leaving the decision of unit up to the student and/or class discussion. Pros, cons?

What you’re missing is that Stanley only has the shorter shovel for about thirty seconds, after which Zero comes and grabs it from him. So Stanley ends up with a regular shovel and Zero has the short one. Therefore the question is correctly phrased.

My point was that if you only read/hear the question without seeing the video (or can’t see the video, not to bring up the fact of visually impaired students), there is ambiguity. It really does sound like Stanley has the short shovel and it’s not clear that he loses it before he gets to use it. And in fact, doesn’t he later wind up with that shovel as a gift from X-Ray?

This is, of course, a minor quibble, but my viewpoint was properly captured by Jonas despite my carelessly omitting the issue of being video- or visually-impaired. Ideally, our written problem statements are consistent with the available visual evidence and there is mutual support between them.

Like any other sort of problem-situation ambiguity, it’s possible to see strengths in not being consistent, clear, well-posed, etc., as well as weaknesses. I MIGHT have been the only person on the planet to wonder what was up (and I’ve actually seen the movie a couple of times, which could be what’s sticking in my craw a bit), but my experience is that if ANYONE sees something as unclear, s/he’s probably not alone.

All that said, I enjoyed lurking on Math 2.0 last night. Most of the ground seemed familiar, but it’s very fertile ground. Frankly, Dan, graduate school will be mostly a waste of time for you. You’re already so far ahead of the thinking of so many mathematics teachers and, dare I say it? mathematics teacher-educators that I wonder if what you’re going to be exposed to and expected to conform to in a doctoral program will improve or dull your mind. Maybe that’s unfair to Stanford, or merely reflective of my own ambivalent relationship with doctoral programs and academia. And perhaps also part of my fond wish that more folks with really great, original minds just forego the rigidity of traditional Ph.D programs if at all possible and carve out their own ground, establish legitimacy through the high quality of their work (as you are CLEARLY well on your way to doing), and let the paper chasers do what seems to pass for establishing their bona fides as insiders who alternately sneer at and quake from fear of originals and iconoclasts.

I feel the conversion from volume to weight does help make the impact more tangible. When considering the effort (work) dig the hole consider the amount of work it takes to lift the last shovel fulls 5 feet (plus 5 foot pile height??) versus the first few shovel fulls that just need to be moved aside. The last item I think is interesting is the weight of the soil volume is highly dependent on moisture. Imagine a problem where it is raining and adding weight to the soil.

WRT the penultimate paragraph of (Dan in grad school), I thought the same thing when I read that you were pursuing a PhD at Stanford. Let me put it this way: “Commanding a starship is your first, best destiny; anything else is a waste of material.” Well, not a waste, but if I could wave a magic wand I’d have you at Stanford and at the same time teaching high school (some place that doesn’t fire you every year).

FWIW, these last six years have left me a bit dazed, and a sabbatical — any kind of sabbatical — seems appropriate to me. A sabbatical studying education at a university of repute seems even better. I could be herding sheep in the Andes.

For WCYDWT desk: and here I am looking for ideas because this bit of information has been on my mind since I read this (in Julia Witty’s piece “BP’s Deep Secrets”, Sept-Oct Mother Jones). Read, marvel, and think.
from the data stream recorded in the ocean near Hawaii…”The animations show spinner dolphins gathering in a circle of 16 or 28 animals, always an even number, each dolphin paired with another, the pairs arranged in an echelon formation: one animal slightly above and ahead of the next, while maintaining about three feet of separation. A perimeter of roughly 300 feet is precisely maintained as the dolphins swim in an undulating circle, trapping the fish inside the net of their swimming bodies.
“One after another, in fixed sequence, two dolphin pairs directly opposite each other dart into the ball of fish to feed. As they return to the circle, four more follow. And so on. The action is extremely fast, the dolphins darting in to feed at a rate of about 1.25 prey per minute, all while swimming and circling in their roller-coaster pattern. After five minutes below, each pair has engaged in four feeding dashes, and the dolphins simultaneously surface to breathe. They typically grab only one or two quick breaths before diving, repeating the underwater rodeo over and over throughout the night without rest.”
This occurs at the DSL (deep scattering layer, the twilight area between surface light and darkness) so no film. But Julia Witty describes a picture easy to visualize. Is mathematics involved if the dolphins can’t articulate what’s going on? Is the mathematics all in our heads? These are questions that mathematics has struggled with for hundreds of years.

I used this in my physics class today to talk about unit conversions and problem solving techniques? My favorite question that the students came up with is how long will it take until they run out of room to dig the holes? I printed them off a screen shot of holes at the begining of the clip to help them out. Thanks again!